Try using select instead of sleep - at least on linux it does not rely on ALRM, it's a single system call.

#!/usr/bin/perl

use strict; # https://perlmonks.org/?node_id=11111522
use warnings; # borrowed from choroba
use feature qw{ say };

local $SIG{USR1} = sub 
  {
  say STDERR "Signal 1 caught.";
  };

local $SIG{USR2} = sub 
  {
  say STDERR "Bye!";
  exit
  };

my $sum = 0;
while (1) 
  {
  $sum += $_  for 1 .. 100;
  say $sum;

  say STDERR scalar localtime, " Sleeping...";
  select undef, undef, undef, 10;

  say STDERR scalar localtime, " Ready!";
  }
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I don't have a Solaris to test on, but this works as expected on my linux system.